Stationarity Conditions for MA(q) and AR(p) Processes

Archit Vora

Sequence and Series

Convergent Sequence

1/2, 2/3, 3/4, . . . , n/(n+1)

Divergent Sequence

3, 9, 27, . . . . , 3^n

Series => Partial Sum of sequence

Convergent Series => if sum converges

 Convergence Test

  • Integral Test
  • Comparison Test
  • Limit comparison test
  • Alternating Series Test
  • Ratio test
  • Root test
Geometric Series

  • a, ar, ar^2, . . . , ar^n
  • Convergent if r < 1
 Representing function as (geometric) series

seriesRepresntation

Backward shift operator

  • B^kX(t) = X(t – k)

backOp

Invertibility

  • Two models have same ACF
  • Given ACF how to find out the model
  • We will go for model that is invertible
  • We can invert MA(1) into AR(∞)
  • Inverting is basically act of expanding function in geometric series
  • It is possible when growth r<1
  • Out of two models only one satisfies this condition
  • We will select that model given ACF

Conditions for Invertibility[MA(q)] and Stationarity [AR(p)]

dual

How to check if series is both invertible and stationary

  • Check AR(p) polynomial for stationarity
  • Check MA(q) polynomial for invertibility
  • Both should hold

Reference

https://www.coursera.org/learn/practical-time-series-analysis/exam/ITocA/series-backward-shift-operator-invertibility-and-duality

 

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